Heavy tail distribution - Weibull

Question

I know that the Weibull distribution exhibits subexponential heavy-tailed behavior when the shape parameter is < 1. I need to demonstrate this using the limit definition of a heavy tailed distribution:

for all

How do I incorporate the cumulative distribution function (CDF) or any other equation characteristic of the Weibull distribution to prove that this limit holds?

Answer 1

The CDF of the Weibull distribution is 1 - exp(-(x/lambda)^k) = P(X <= x).

So

P(X > x) = 1 - CDF = exp(-(x/lambda)^k),

and

lim exp(lambda * x) * P(X > x) = lim exp(lambda x) * exp( - (x/lambda)^k)
                               = lim exp(lambda x - x^k/lambda^k)

Since k<1, and x is large, and lambda>0, lambda x grows large faster than x^k/lambda^k (the monomial with the greater exponent wins). In other words, the lambda x term dominates the x^k/lambda^k term. So lambda x - x^k/lambda^k is large and positive.

Thus, the limit goes to infinity.